Leaving aside why would Great Britain grant independece to the Falkland Islands in the 1870s, so in the era of colonial expansionist hysteria in its zenith, I see two small problems here:
- a population of eight thousand people, even the richest ones, couldn't support more than few small patrol boats. In the real world it's military - and I mean whole military: land, sea and air combined - wouldn't exceed 100 men - ok. it's a rich country so let's say 200 which is very understrength infantry company (without any weapons heavier than a machine gun) plus 2 or 3 patrol boats of some 20-30 meters and a single aircraft maybe the size of Beech King Air - and that's a very optimistic assumption.
- You wrote that nominal GDP is 53 billion USD but GDP per capita is 65 thousand USD. Sorry but math doesn't work here. 65.000$ x 8.585 (population) = 558.025.000$ so around 1% of the nominal GDP You mentioned.

Great to see that someone else on this forum thinks the Falkland Islands is a good place to start an AU project!

I have two variants of a Falkland Islands AU in development on this forum;

1: a version that is still largely a dependency of Great Britain but develops its own military capacities in the years surrounding the 1982 Argentine Invasion that allow some interesting 'what-ifs' to be explored. And;

2: a full blown AU nation of considerable economic and military power that gains independence as a Commonwealth Dominion in the 1890s and plays a substantial role in the geopolitical events of the 20th Century. This Au nation encompasses a significant portion of the South American land mass, in addition to the falklands themselves.

Check out the following links to threads i've already got in the works.